Problem: Aliaa and Zhang Li are tennis-playing robots capable of placing shots with superhuman precision. Aliaa is about to hit its next shot to the far corner of the court. It (Aliaa) knows that the distance between itself and Zhang Li is $520\,\text{cm}$, and it knows that the angle between Zhang Li and the far corner is $53^\circ$. Similarly, Zhang Li knows that the angle between Aliaa and the far corner is $100^\circ$. What distance should Aliaa hit the ball so that it lands perfectly in the far corner of the court? Do not round during your calculations. Round your final answer to the nearest centimeter.
Answer: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $BC=d$. Because the interior angles of a triangle add to $180^\circ$, we know that $\angle C=27^\circ$. $A$ $B$ $C$ $53^\circ$ $100^\circ$ $27^\circ$ $520\text{ cm}$ $d$ Since we are given one side length and all angle measures, we can use the law of sines. Using the law of sines $\begin{aligned} \dfrac{\sin(C)}{AB}&=\dfrac{\sin(A)}{BC}\\\\ \dfrac{\sin(27^\circ)}{520} &= \dfrac{\sin(100^\circ)}{d} \gray{\text{Substitute}} \\\\ d \cdot \sin(27^\circ) &= 520 \cdot \sin(100^\circ) \\\\ d &= \dfrac{520 \cdot \sin(100^\circ) }{\sin(27^\circ) } \\\\ d &\approx 1128 \,\text{cm} \end{aligned}$ The answer Aliaa should hit the ball $1128 \,\text{cm}$.